TAILIEUCHUNG - SAS/ETS 9.22 User's Guide 108

SAS/Ets User's Guide 108. Provides detailed reference material for using SAS/ETS software and guides you through the analysis and forecasting of features such as univariate and multivariate time series, cross-sectional time series, seasonal adjustments, multiequational nonlinear models, discrete choice models, limited dependent variable models, portfolio analysis, and generation of financial reports, with introductory and advanced examples for each procedure. You can also find complete information about two easy-to-use point-and-click applications: the Time Series Forecasting System, for automatic and interactive time series modeling and forecasting, and the Investment Analysis System, for time-value of money analysis of a variety of investments | 1062 F Chapter 18 The MODEL Procedure Estimate the true parameter vector 00 by the value of 0 that minimizes S 0 V nm 0 V-1 nm 0 n where V Cov nm 00 nm 0o 0 The parameter vector that minimizes this objective function is the GMM estimator. GMM estimation is requested in the FIT statement with the GMM option. The variance of the moment functions V can be expressed as V n n 0 Eg zJ E s zJ t 1 s 1 X X E ft 0zt g0Zs 0 t 1s 1 nsn where S J is estimated as S n 1 XX q yt xt 0 0zt q ys x5 0 0z5 0 n 1 1t 1 Note that Sn is a gkxgk matrix. Because Var Sn does not decrease with increasing n you consider estimators of S0n of the form S n l n n 1 X r n 1 T _ wW n rD S n r n P q yt Xt 0 0Zt q yt_r x _T 0 z -r 0 t 0 t 1 r Sn -r 0 T 0 where l n is a scalar function that computes the bandwidth parameter w - is a scalar valued kernel and the diagonal matrix D is used for a small sample degrees of freedom correction Gallant 1987 . The initial 0 used for the estimation of Sn is obtained from a 2SLS estimation of the system. The degrees of freedom correction is handled by the VARDEF option as it is for the S matrix estimation. The following kernels are supported by PROC MODEL. They are listed with their default bandwidth functions. Estimation Methods F 1063 Bartlett KERNEL BART w x Í1 X 1x1 D 1 0 otherwise l n n1 3 Parzen KERNEL PARZEN w x l n 1 6 x 2 6 x 3 2 1 - x 3 0 n1 5 0 X 2 2 x 1 otherwise Quadratic spectral KERNEL QS w x l n 25 sin 6nx 5 12n2x2 6nx 5 cos 6nx 5 -n1 5 2 1064 F Chapter 18 The MODEL Procedure Figure Kernels for Smoothing Details of the properties of these and other kernels are given in Andrews 1991 . Kernels are selected with the KERNEL option KERNEL PARZEN is the default. The general form of the KERNEL option is KERNEL PARZEN QS BART c e where the e 0 and c 0 are used to compute the bandwidth parameter as l n cne The bias of the standard error estimates increases for large bandwidth parameters. A warning message is produced for bandwidth parameters greater than

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